Scherrer Equation Calculator

Calculator Inputs

Use the engineering form below to estimate coherent diffraction domain size from X-ray diffraction peak broadening.

Shape factor (K)

Common values range near 0.89 to 0.94.

Wavelength value

Enter the X-ray wavelength used for the scan.

Wavelength unit

Å is common for Cu Kα radiation.

Peak position (2θ)

Enter the diffraction peak center as 2θ.

Angle unit

The calculator converts 2θ to θ automatically.

Observed FWHM

Enter the measured peak full width at half maximum.

Instrumental FWHM

Use 0 if you do not have instrument broadening data.

FWHM unit

Scherrer broadening is finally used in radians.

Decimal places

Choose your preferred output precision.

Apply instrumental broadening correction

Plotly Graph

This graph shows how estimated crystallite size changes as FWHM broadening changes near your current input range.

Example Data Table

These sample rows show how the Scherrer equation responds to different diffraction conditions.

Case	K	Wavelength (Å)	2θ (°)	Observed FWHM (°)	Instrumental FWHM (°)	Corrected FWHM (°)	Estimated Size (nm)
Example 1	0.90	1.5406	36.5	0.24	0.06	0.2324	36.00
Example 2	0.94	1.5406	44.7	0.31	0.08	0.2995	29.95
Example 3	0.89	0.7093	28.2	0.18	0.03	0.1775	21.01

Formula Used

Scherrer equation: D = (K × λ) / (β × cos θ)

Instrument-corrected broadening: β = √(β_obs² − β_inst²)

Here, D is crystallite size, K is shape factor, λ is wavelength, β is peak broadening in radians, and θ is the Bragg angle.

The equation estimates coherent diffraction domain size. It does not always equal the physical particle size because strain, defects, and agglomeration can affect peaks.

How to Use This Calculator

Enter the Scherrer shape factor for your material or crystal habit.
Provide the X-ray wavelength in Å or nm.
Enter the diffraction peak position as 2θ and choose its unit.
Input the observed FWHM from your fitted diffraction peak.
Enter instrumental FWHM if known and enable correction.
Choose the number of decimal places you want.
Press the calculate button to view size, corrected broadening, angle terms, and graph output.
Use the CSV and PDF buttons to save the calculation summary.

Frequently Asked Questions

1. What does the Scherrer equation calculate?

It estimates crystallite size from X-ray diffraction peak broadening. The result reflects coherent diffraction domain size, not necessarily the full particle diameter seen in microscopy.

2. Why must FWHM be in radians?

The equation uses angular broadening in radians. This calculator accepts degrees or radians, then converts values internally before applying the size formula.

3. What is a common shape factor value?

A common default is 0.90. Some analyses use values near 0.89 or 0.94, depending on crystallite shape and peak definition.

4. Should I correct for instrumental broadening?

Yes, when reliable instrument broadening data exists. Without correction, the calculated size may appear smaller because measured peak width includes both sample and instrument effects.

5. Does this output equal particle size?

Not always. The equation estimates coherent crystalline domain size. Actual particles can be larger if they contain multiple domains or defects.

6. Can strain affect the result?

Yes. Microstrain broadens diffraction peaks and can bias Scherrer estimates. For strained materials, methods like Williamson-Hall analysis are often more informative.

7. Which wavelength should I enter?

Enter the X-ray wavelength used during your scan. For example, Cu Kα radiation is often entered as 1.5406 Å.

8. Why does larger FWHM give smaller size?

The equation places broadening in the denominator. As peak width increases, the estimated crystallite size decreases, assuming all other inputs stay unchanged.